Big Idea:
Students will write formal proofs showing how triangles are congruent when two corresponding sides and the included angle are congruent.

In today's Do Now, students are asked three questions that can be answered using the given diagrams. The first question asks the students to identify what they know about the two triangles. For example, they might say, "side AB is congruent to side AD and side AC is congruent to side DF."

The second question asks students what information is needed to prove the two triangles are congruent. There are multiple answers to this question. For example, side BC would have to be congruent to side EF or angle BAC would need to be congruent to angle EDF. In general, students will answer this question depending on what they recall from a previous lesson on triangle congruence. The answer to the final question on the Do Now depends on how students answer question two.

When we go over the answers, I plan to call on a student who wrote that the triangles will be congruent based on the side-side-side postulate. I have been looking for a "volunteer" as I circulated observing students work. After the student presents, I will ask if there is another way to prove the two triangles are congruent. Some students will remember the side-angle-side postulate, which we looked at briefly in an earlier lesson. If none of the students recall this postulate, I have them look back to the graphic organizer they created in the lesson on triangle congruence. This discussion leads us into a more in depth investigation in the Mini-Lesson.

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The key concept we review in the Mini-Lesson is the term “included angle.” This terminology is often new to the students and they sometimes have difficulty identifying the included angle. We practice identifying the included angle between two sides in the triangles from the Do Now.

I ask students to identify the included angle between sides AB and AC. Then they are asked to identify the included angle between sides ED and FD. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Conversly, if two triangles are congruent, then the two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. Using this statement, I have the students list other pairs of congruent sides and angles that can be used to prove the two triangles from the Do Now are congruent. In the activity section, students practice writing proofs involving the angle-side-angle postulate.

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In the Activity, students work independently to write two column proofs involving the side-angle-side postulate. I have the students take the first few minutes to look at the given statements and label the information on the diagrams. Then they brainstorm about what information they will need to write the proof. They use information explored in previous lessons to write the proofs.

At this point in the unit, students should be able to work on their own to write the proofs. If students have difficulty on their own, I pair up two students to help each other.

After about 20 minutes, I call on a student to present his or her proof. We critique the student’s reasoning and fix any misconceptions or errors. Sometimes students skip steps when writing proofs. Depending on the level of the class or the students, I accept different answers when necessary. I have provided some examples of Student Work.

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At the end of the lesson, students complete an Exit Ticket, which contains a proof involving SAS. Students can write a paragraph proof or a two-column proof. They work independently and I collect the exit ticket to see how students are doing with proofs. This exit ticket can be used as a formative assessment, which is ungraded or as a quiz that counts towards their class grade.